Question 1

Which of the following scenarios is most appropriately modeled by a Poisson distribution?

Accepted Answer

The number of cars passing through a toll booth in one hour.

Question 2

Given $$X \sim 	ext{Poi}(4)$$, what is the probability that exactly 2 plates are broken next week?

Accepted Answer

$$P(X=2) = \dfrac{e$$^{-4}$$ 4^2$}{2!} = 0.1465$$

Question 3

For a Poisson random variable $$X$$ with mean $$\mu$$, which formula gives the probability of observing $$x$$ events?

Accepted Answer

$$P(X=x) = \dfrac{e$$^{-\mu}$$ \mu^x}{x!}$$

Question 4

At a hospital, babies are born at a rate of 20 per hour. What is the expected number of babies born in 15 minutes?

Accepted Answer

$$\mu = 5$$

Question 5

If $$X \sim 	ext{Poi}(4)$$, what is the probability that at most 1 plate is broken next week?

Accepted Answer

$$P(X \leq 1) = \dfrac{e$$^{-4}$$ 4^0$}{0!} + \dfrac{e$$^{-4}$$ 4^1$}{1!} = 0.0916$$

Question 6

Suppose $$X \sim 	ext{Poi}(3.5). $$What is $$P(X \u003c 6)$$?

Accepted Answer

$$P(X \u003c 6) = 0.9858$$

Question 7

A port city has an average of 8 tankers arriving per day. What is the probability that there are fewer than 3 tankers tomorrow?

Accepted Answer

$$P(X \u003c 3) = P(0) + P(1) + P(2) = 0.0138$$

Question 8

Which of the following is NOT an assumption of the Poisson distribution?

Accepted Answer

The probability of an event depends on previous events.

Question 9

If the average number of server crashes per year at UKZN is 9, what is the probability that exactly 11 servers crash this year?

Accepted Answer

$$P(X=11) = \dfrac{e$$^{-9}$$ 9$$^{11}$$}{11!}$$

Question 10

A random sample of 9,000,000 airplanes is selected. If the probability that a commercial airplane is involved in a fatal accident is $$0.0000004$$, which distribution can be used to approximate the number of airplanes involved in fatal accidents?

Accepted Answer

Poisson distribution with $$\mu = 3.6$$

Question 11

If $$X \sim 	ext{Poi}(\mu)$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = \mu$$

Question 12

A customer arrives at a takeaway every 3 minutes, independently. What is the expected number of customers in 4 hours?

Accepted Answer

$$80$$

Question 13

Which of the following best describes the variance of a Poisson random variable $$X \sim 	ext{Poi}(\mu)$$?

Accepted Answer

$$	ext{Var}(X) = \mu$$

Question 14

If $$X \sim 	ext{Poi}(4)$$, what is the probability that more than 3 plates are broken next week?

Accepted Answer

$$P(X \u003e 3) = 1 - [\dfrac{e$$^{-4}$$ 4^0$}{0!} + \dfrac{e$$^{-4}$$ 4^1$}{1!} + \dfrac{e$$^{-4}$$ 4^2$}{2!} + \dfrac{e$$^{-4}$$ 4^3$}{3!}]$$

Question 15

Which of the following is a key difference between the Poisson and Binomial distributions?

Accepted Answer

Poisson models the number of events in a fixed interval, Binomial models the number of successes in a fixed number of trials.

Question 16

If the average number of tankers in a port in half a day is 4, what is the probability that at least 2 but less than 6 tankers are present?

Accepted Answer

$$P(2 \leq X \u003c 6) = P(X=2) + P(X=3) + P(X=4) + P(X=5)$$

Question 17

Which formula would you use to find the probability that no events occur in a Poisson process with mean $$\mu$$?

Accepted Answer

$$P(X=0) = e$$^{-\mu}$$

Poisson Distribution: Concepts, Formulas, and Applications

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